I Kets and vectors

Hi. I have read that when working in 3-D the following kets | x > and | p > are not vectors in 3-D. If that is correct what are they ? I know | ψ > is an abstract vector but I thought | x > and | p > would be 3-D vectors in the position and momentum representation ?
Thanks

They are vectors in an infinite-dimensional space. It is natural to feel that ##|x\rangle## should be in a 3d space because ##x##, without the enclosing symbols, is a coordinate in 3D space. But ##|x\rangle## is not a coordinate or any other type of number. In the position representation it is the Dirac delta function ##\delta_x##, which is an element of a space of functions, which - like most function spaces - is infinite-dimensional.